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Properties

Label 2-99-9.7-c1-0-9
Degree $2$
Conductor $99$
Sign $-0.173 + 0.984i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s − 1.73i·3-s + (−0.999 − 1.73i)4-s + (1 + 1.73i)5-s + (−2.99 − 1.73i)6-s + (−2 + 3.46i)7-s − 2.99·9-s + 3.99·10-s + (−0.5 + 0.866i)11-s + (−2.99 + 1.73i)12-s + (−2 − 3.46i)13-s + (3.99 + 6.92i)14-s + (2.99 − 1.73i)15-s + (1.99 − 3.46i)16-s + 4·17-s + (−2.99 + 5.19i)18-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s − 0.999i·3-s + (−0.499 − 0.866i)4-s + (0.447 + 0.774i)5-s + (−1.22 − 0.707i)6-s + (−0.755 + 1.30i)7-s − 0.999·9-s + 1.26·10-s + (−0.150 + 0.261i)11-s + (−0.866 + 0.499i)12-s + (−0.554 − 0.960i)13-s + (1.06 + 1.85i)14-s + (0.774 − 0.447i)15-s + (0.499 − 0.866i)16-s + 0.970·17-s + (−0.707 + 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880914 - 1.04983i\)
\(L(\frac12)\) \(\approx\) \(0.880914 - 1.04983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.09469991185346332045386583260, −12.62844252177810186401158605939, −11.85230104161715369730063398338, −10.66881913959611422418747613477, −9.653262976223268069337490628147, −8.017926436543149103413944378313, −6.49567444727210546409205802887, −5.41240376141583273152091738853, −3.06110036737768672608680415369, −2.29631348731773603655789539034, 3.87777778052902744490531742535, 4.81185525673423608439381107896, 6.00774878416193081244758170367, 7.19537421954810871040449130182, 8.639877389633739775295062926979, 9.834799830741763466408613400424, 10.74931166116132776070400334915, 12.53130220306551015781220229495, 13.58480207612691748941818361235, 14.23418899392058021219652899344

Graph of the $Z$-function along the critical line